You're retrieving the date value from a text field on the form in the format "m/d/yyyy." That date value, as presented, is not in the format required for writing back into a SharePoint list. You have to write dates to SharePoint in the ISO 8601 format such as "yyyy-mm-ddTh:mm:ss-00:00." There is a function in the thread at http://spservices.codeplex.com/discussions/349356 that will convert a JavaScript date object to the format you need. Of course, it requires a JavaScript date object to start. To convert your existing variable to a JavaScript Date object, you can use...
var dateComp = new Date($('h3:contains("Date Completed")').closest('td').next('td').text());
...Then pass 'dateComp' to the ISO date conversion function. The return from that function should be what you need to write to your list.
Geoff